

Preprint 7/2008
Compensated Compactness, Separately convex Functions and Interpolatory Estimates between Riesz Transforms and Haar Projections
Jihoon Lee, Paul F. X. Müller, and Stefan Müller
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Submission date: 31. Jan. 2008
Pages: 52
published in: Communications in partial differential equations, 36 (2011) 4, p. 547-601
DOI number (of the published article): 10.1080/03605301003793382
Bibtex
MSC-Numbers: 49J45, 42C15, 35B35
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Abstract:
In this work we prove sharp interpolatory estimates
that exhibit a new link between Riesz transforms and directional projections of the Haar system in
To a given direction
we let
be the orthogonal projection onto the span of those Haar functions
that oscillate along the coordinates
When
the identity operator and the Riesz transform
provide a logarithmically convex estimate for
the
norm of
see Theorem 1.1.
Apart from its intrinsic interest Theorem 1.1
has direct applications to variational integrals, the theory of
compensated compactness, Young measures, and to the relation between
rank one and quasi convex functions.
In particular we exploit our Theorem 1.1
in the course of proving a conjecture of L. Tartar
on semi-continuity of separately convex integrands;
see Theorem 1.5.